Wow, that's an old answer that /u/silverappleyard linked. It still stands up in its basic logical points, though /u/Tiako 's note about the fraught relations between Roman Egypt and Meroë (Nubia) is well taken. If the Nile was the highway past the Sahara, Meroë was the roadblock. Evidence is that, in field battles, the Romans likely prevailed; however, raiding north and south created a situation where any gain was short-lived. It helped that Meroë had a powerful ironworking industry that produced high-grade goods, and wealthy trade connections because of the same, which brought items from further south and west into the orbit of Egypt. We still can't read Meroitic reliably (despite having enough writing to know the syllabary) but the diplomatic settlement with Apedemek and successors was far more cost-effective.
The Periplus of the Erythrean Sea talks about Red Sea destinations and a bit about Rhapta, the somewhat mysterious large East African port whose location we don't exactly know (likely because another city is atop it now), but these areas were not likely to face Roman invasion for the same reason they did not face Islamic invasion--taking and holding those areas amphibiously was not a profitable or pragmatic long-term strategy, when greater benefit came from controlling the crossing point of Egypt itself. The conversion of Ezana of Aksum (today in Eritrea) to Christianity was based on influence from Egypt, and that conversion made Adulis a major friendly port for Roman Egypt through the 4th and 5th centuries. So it wasn't exactly necessary to establish dominion there to get the benefits, and they didn't need the land--at least not land in those latitudes.
As far as what the Romans knew, Herotodus, Strabo, and others give a pretty good idea of what information was in circulation. But we have little evidence of Roman travelers going farther afield than the upper Nile, or at least doing so and then writing accounts of those travels. Paul Halsall has collected some of the extant accounts of 'inner Africa' into his Internet History Sourcebook at Fordham. One thing that's notable is the incidence of disease and the higher morbidity rate further south doesn't make much of an appearance, which makes one wonder if people understood that link. That's a question beyond my ken, though.
[edit: clarified para 3 a bit]
While there’s always more to say on the subject, you might be interested in these previous answers: